Question

Simplify: $\frac{\sqrt{3}\times 9^{\frac{1}{2}}}{81}$.

Answer 1

$\frac{\sqrt{3}\times 9^{\frac{1}{2}}}{81}$ $=\frac{3^{\frac{1}{2}}\times {\left(3^2\right)}^{\frac{1}{2}}}{3^4}$

$=3^{\frac{1}{2}+\frac{2}{2}-4}$ (using ${\left(a^m\right)}^n=a^{mn}$ and $\frac{1}{a^m}=a^{-m}$)

$=3^{(\frac{1}{2}+1-4)}$ $=3^{(\frac{1}{2}-3)}$ $=3^{\frac{-5}{2}}$ $=\frac{1}{3^{\left(\frac{5}{2}\right)}}$

$=\frac{1}{\sqrt{243}}=\frac{1}{9\sqrt{3}}$.

Therefore, $\frac{1}{9\sqrt{3}}$ is the simplified form.